A silly question, but I don't see the answer.
This question is from Rudin's Functional Analysis, Chapter 10, Exercise 1a).

It's obvious that $y$ has a left inverse as $((xy)^{-1}x)y=e$, where $e$ is the unit of the algebra. However, starting with $y((xy)^{-1}x)$, I cannot proceed further. Does this suggest that I need to 'venture beyond' the multiplicative group structure, maybe introduce ideas from the ring structure on the Banach Algebra? ( I'm thinking of something analogous to the proof where $e-xy$ is invertible if and only if $e-yx$ is invertible. There, if $z$ is the inverse of $e-xy$, then, $e+yzx$ is the inverse of $e-yx$.)